Mathematical Modelling

Mathematical Modeling

Xavier J.R. Avula , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

X Concluding Remarks

In this article, mathematical modeling concepts in relation to physical sciences, engineering, and technology have been surveyed. In recent years, mathematical modeling has pervaded all branches of knowledge, bringing forth greater understanding of processes under investigation. In engineering and technology it provides the analytical basis for design and control in which predictions can be confidently made without spending valuable resources of money and effort.

Successful applications of mathematical modeling techniques in engineering sciences have led the way to extend the techniques to more exotic areas of inquiry, like nanotechnology, nuclear-reactor engineering, material science, environment, weather prediction, biological processes, space sciences, cosmology, and also social sciences. Although the general philosophy of modeling in these new areas remains the same as discussed in this article, the simulation procedures and validation criteria are different and dependent on the types of models and the disciplines they belong to.

Mathematical modeling is a vast, multidisciplinary field that pleads to engage the interest and dediation of engineers, scientists and mathematicians to solve the problems facing the humankind. A significant development in the mathematical modeling activity is the availability of very-high-speed computers, which can solve a variety of complex models. In spite of all the advances in empirical knowledge, solution techniques, and computer assistance, it must be noted that human intelligence, experience, and intuition still play a significant role in mathematical modeling.

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28th European Symposium on Computer Aided Process Engineering

Ivan Červeňanský , ... Jozef Markoš , in Computer Aided Chemical Engineering, 2018

5 Conclusions

Mathematical modeling has currently an irreplaceable position in process intensification but its application in biocatalytic production of natural flavors and fragrances is difficult mainly because of very specific kinetics of the biotransformation process. Such specific kinetics cannot be easily implemented into universal process simulators which are often used for the search of optimal working conditions. Also, mathematical models of membrane separation processes are not developed in process simulators. Therefore, in these cases the only way of using mathematical modeling for the search of optimal working conditions is creating a custom program tool. In this work, such a tool has been created for the bioproduction of PEA, which is an important natural flavor. The developed program tool combines mathematical models of bioproduction, microfiltration, membrane extraction and distillation which can work in batch configuration or as one hybrid system with continuous product removal. This tool can be used to compare batch and hybrid systems or to study and intensify PEA bioproduction.

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23rd European Symposium on Computer Aided Process Engineering

Tekena Fubara , ... Aidong Yang , in Computer Aided Chemical Engineering, 2013

Abstract

Mathematical modelling and optimization of the natural gas based Distributed Energy Supply System (DESS), both at the building level and the overall energy supply network level was carried out for three types of micro-CHP – solid oxide fuel cells, Stirling engines, internal combustion engines – and for two operating strategies – cost-driven and primary energy-driven. The modelling framework has particularly allowed the quantification of the impact of micro-CHP on the total primary energy consumption at the whole network level. The result of a case study based on the UK reveals the range of the overall reduction in primary energy usage and central power plant capacity requirement and the range of the increase in natural gas supply to homes. The economic analysis shows that the coupling of different technologies, sizes of the CHP engine, and the operating strategies gives rise to a wide range of payback time.

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26th European Symposium on Computer Aided Process Engineering

Tannaz Tajsoleiman , ... Ulrich Krühne , in Computer Aided Chemical Engineering, 2016

5 Conclusions

Mathematical modelling and CFD simulation are two techniques which can be employed in the design of an optimal scaffold micro-architecture for cell culturing processes. In this work, a geometry optimization routine was proposed and applied for one of the common types of scaffold for cartilage regeneration in a perfusion bioreactor to indicate the potential of these tools. By applying this routine, various geometries were studied by moving 24 points to create new geometries in a numerical environment in order to reach the optimal geometry with high culture efficiency. In this perspective, modelling and simulation were used to predict the cell distribution in different geometries and environmental conditions without performing any experiments during the culture time. The developed simulation was built on a set of equations that present the correlation between the cell growth rate, glucose concentration and local shear stress level. In future studies, this routine can be implemented for other types of scaffold by changing the number of points and their freedom of movement.

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Batch Processing

Narses Barona , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

VII.C System Modeling

Mathematical modeling of systems for which characteristic variables are time-dependent only and not space-dependent is done by ordinary differential equations (ODEs). The situation is found in a nearly well-mixed batch reactor. There one may find differences in temperature or concentrations from one site to another due to imperfect mixing. When space changes are not important to the model, the process variables can be approximated by means of lumped parameter models (LPMs). When the space change of certain variables is important, the description of these process variables must be in terms of partial differential equations (PDEs). The changes of these variables may be interpreted only by distributed parameter models (DPMs). An illustration is encountered in a system of large viscosity where the temperature changes along the direction of heat flow are as important as the time changes in determining product quality.

Algebraic equations (AEs) describe relations among variables of the system which are independent of time or space changes. They represent variables not related by material energy or momentum balances. They may characterize physicochemical or other type of relationships between physically independent portions of the system, as the vapor and liquid spaces of a reaction vessel.

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30th European Symposium on Computer Aided Process Engineering

S. Elmisaoui , ... J.M. Ghidaglia , in Computer Aided Chemical Engineering, 2020

Abstract

Mathematical modelling and numerical simulations are widely used in the petrochemical industries for operator training, design, and process optimization. However, there is a lack of rigorous numerical modelling and simulations in the phosphate fertilizer industry. There exist many challenges in the production systems of phosphate fertilizers including (i) multiphase flows in the system involving liquids, solids or gases, (ii) particles with different size distributions, and (iii) dynamic variations in the physical properties including rheology and thermodynamic behavior.

In the current study, using well-established techniques from computational fluid dynamics, we develop a model for the numerical simulation of multiphase flows in a conditioner operation unit used in the phosphate fertilizers facilities. The proposed model deals with the first step in the process, consisting on the preneutralizer, and it uses the Reynolds-averaged Navier-Stokes equations for modeling turbulent flow in the system. The preneutralization consist on a partial reaction between ammonia and phosphoric acid. Numerical results are presented for several scenarios and we particularly show the importance of the baffles on eliminating vortices, and also their effect on the different hydrodynamic performance criteria. This approach can be used to describe the behavior of reaction systems within this type of industrial plants and the analysis of the other sub-process of the fertilizer plant is an ongoing work.

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Effects of Excessive Water Intake on Body-Fluid Homeostasis and the Cardiovascular System — A Computer Simulation

Y. Zhang , ... V. Gupta , in Emerging Trends in Applications and Infrastructures for Computational Biology, Bioinformatics, and Systems Biology, 2016

Abstract

Mathematical modeling of the cardiovascular system is useful in quantifying system-level responses to external and internal perturbations, which are challenging to achieve in clinical studies. Most existing cardiovascular models are closed-loop systems and do not take into account the fluid and electrolyte exchanges and the body-fluid homeostasis. We propose a new system-level simulation model that modifies and couples an open-source cardiovascular model with a renal system model. The new model has the capability to simulate both the short- and long-term responses of the cardiovascular system and the body-fluid homeostasis due to changes in the amount of fluid intake. In the present study, we apply the model to investigate the short- and long-term changes in the human cardiovascular and renal systems, in responses to excessive fluid intake. For the short-term simulations, an instant but mild increase in mean arterial pressure is found, while the heart rate decreases slightly, after an ingestion of 500   mL of water. The results agree reasonably well with data from clinical studies. Simulation results also show that a prolonged elevation of daily fluid intake leads to chronic adaptations of system parameters, such as the blood pressure and hormonal levels. These adaptations cause an increase in the left ventricular stroke work during each cardiac cycle, which may have implications on the chronic remodeling of the myocardial structures and its mechanical functions. These chronic overload remodeling may eventually lead to pathological cardiovascular conditions, such as cardiac arrhythmias and congestive heart failure.

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28th European Symposium on Computer Aided Process Engineering

Marcus Fechtner , ... Achim Kienle , in Computer Aided Chemical Engineering, 2018

1 Introduction

Mathematical modeling of spatially distributed packed bed adsorbers leads to systems of partial differential equations. An important class of models assume thermodynamic equilibrium between the fluid and the solid phase (Rhee et al., 1986, 1989). Besides explicit isotherms also implicit formulations are quite common to describe the adsorption equilibrium. Typical examples are stoichiometric ion exchange (Tondeur, 1969; Helfferich and Klein, 1970) and the ideal as well as real adsorbed solution theory approaches (Ruthven, 1984). These are computationally much more difficult to treat due to the implicit phase equilibrium. Usually some challenging implicit analytical or numerical differentiation of the equilibrium relations is required to calculate the capacity matrix of the model equations (Kaczmarski and Antos, 1999; Landa et al., 2013). Recently, we developed an approach for the efficient simulation of packed bed adsorbers with implicit adsorption isotherms (Fechtner and Kienle, 2017). The resulting system of partial differential and algebraic implicit equations is reformulated such that explicit differentiation of the isotherm is avoided. Using the method of lines (Schiesser, 1991), the resulting differential algebraic equations (DAE) can be solved simultaneously using standard software. Application was demonstrated for stoichiometric ion exchange with constant solution normality. In this paper, the extension to systems with variable solution normality is presented. This extension allows to consider a large variety of additional processes like salt gradient elution and / or protein separation (Carta and Jungbauer, 2010). In the present paper, application to step gradient elution (Karkov et al., 2013) with stoichiometric ion exchange and linear gradient elution (Gallant et al., 1995) with steric mass action law (Brooks and Cramer, 1992) for protein separation is demonstrated.

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Design Languages

Ian Grout , in Digital Systems Design with FPGAs and CPLDs, 2008

4.7 Mathematical Modeling Tools

Mathematical modeling and simulation tools are increasingly used in designing hardware circuits and systems because they allow fast development and interpretation of the algorithms that the hardware is to implement. A number of mathematical tools exist:

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MATLAB® [21, 22]

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Mathematica [23]

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Modelica [24]

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Maple [25]

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Scilab [26]

As an example of such a tool, consider MATLAB® from The Mathworks Inc. It integrates mathematical computing and data visualization tasks that are underpinned with the tool using its own modeling language. MATLAB® is accompanied with a range of toolboxes, blocksets, and other tools that allow a range of engineering and scientific applications. In such an approach, various ideas can be investigated as part of an overall design process to arrive at a final and optimal solution. The toolboxes and blocksets are utilized for:

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data acquisition

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data analysis and exploration

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visualization and image processing

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algorithm prototyping and development

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modeling and simulation

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programming and application development

Examples of the currently available toolboxes and blocksets are shown in Table 4.1.

Table 4.1. Example toolboxes within MATLAB®

Communications Blockset A blockset that builds on the Simulink® system level design environment for modeling the physical layer of a communication system.
Communications Toolbox A library of MATLAB® functions that supports the design of communication system algorithms and components. It builds on the powerful capabilities of MATLAB® and the Signal Processing Toolbox by providing functions to model the physical layer of a communication system.
Control System Blockset A collection of algorithms that implement common control system design, analysis, and modeling techniques.
Filter Design HDL Coder Filter Design HDL Coder allows for the generation of synthesisable and portable HDL code for fixed-point filters that have been designed using the Filter Design toolbox. Both Verilog® -HDL and VHDL code can be generated. It also automatically creates VHDL and Verilog® -HDL test fixtures/test benches for simulating, testing, and verifying the generated HDL code.
Filter Design Toolbox The Filter Design Toolbox extends the Signal Processing Toolbox. It is a collection of tools that provide techniques for designing, simulating and analysing digital filters with filter architectures and design methods for complex real-time DSP applications
Fuzzy Logic Toolbox Provides a graphical user interface to support the steps involved in fuzzy logic design.
Signal Processing Toolbox A collection of MATLAB® functions that provides a customizable framework for analogue and digital signal processing.
Simulink® An interactive tool for modeling, simulation, and analysis of dynamic, multidomain systems using a graphical, block diagram approach.
Simulink® Fixed Point Simulink® Fixed Point allows for the design of control and signal processing systems using fixed-point arithmetic.
Simulink® HDL Coder Simulink®HDL Coder allows for the generation of synthesisable and portable HDL code from Simulink® models, Stateflow® charts and Embedded MATLAB® code. Both Verilog®-HDL and VHDL code can be generated.
Stateflow® Stateflow® extends Simulink® for developing state machines and flow charts through a design environment. It provides language elements required to describe complex logic in a natural, readable, and understandable form.

Simulink® is commonly used by control system designers and increasingly by electronic circuit designers to model the operation of the required circuit or system in a block diagram format. As an example of this, consider a SISO (single input, single output) closed-loop DC motor control system. Here, speed control is required with no steady-state error. The motor is modelled as a first-order system with a Laplace transform and is controlled by a PI (proportional plus integral) controller. Figure 4.23 shows the motor control system block diagram with a PI controller.

Figure 4.23. Motor control system example with PI control

An example Simulink® model for this system is shown in Figure 4.24.

Figure 4.24. Simulink® model for the motor control system example

Therefore, in this model:

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The motor is modeled as a Laplace transform with the transfer function 1/(1 + 0.1s).

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The proportional gain is 2, and the integral gain is 8 (not optimized).

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This is a high-level behavioral model and does not take into account aspects such as value limits, slew rate, and dead-zones.

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The motor model contains the tachogenerator.

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The command input (required speed) and actual motor speed outputs here are considered to be voltages, and the motor shaft speed uses suitable units (e.g., rads/sec).

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The model uses the built-in Simulink® library blocks, and no design hierarchy has been developed.

The motor model is a simple first-order Laplace transform that models the motor and tachogenerator as a single unit. It was created by monitoring the tachogenerator output voltage to a step change in motor speed command input voltage. This is reasonably representative of the motor reaction to larger step changes in command input, but does not model nonideal characteristics such as a motor dead-zone around a null (zero) command input and the need for a minimum command input voltage required for the motor to react to a command input change.

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Climate Change and Health

A. Haines , in International Encyclopedia of the Social & Behavioral Sciences, 2001

2.5 Mathematical Modeling of Malaria

Mathematical modeling has been applied to the assessment of likely changes in the geographical range of vector-borne diseases such as malaria. One estimate, for example, suggests that approximately 45 percent of the world's population live in zones of potential malaria transmission as defined by current climatic circumstances, and this would increase to around 60 percent towards the end of the next century assuming other relevant factors remain constant (Martens et al. 1995). Highly aggregated models such as the one used in this example are, of necessity, unable to take into account complexity of future changes. Nevertheless, they give a broad indication of the potential magnitude and direction of change and are continually being refined. They suggest that changes in distribution of malaria are likely to occur particularly at the edges of the current distribution, including, for example, mountainous regions in the tropics and subtropics. Estimation of numbers of excess cases and deaths in the twenty-first century as a result of climate change is hampered by our lack of knowledge, for example, about the potential advances in the development of an effective vaccine for malaria, the distribution of impregnated bed nets to reduce transmission, and the trends in the development of resistance of the parasites to drugs used in treatment.

A number of empirical studies in Zimbabwe, Rwanda, and Ethiopia have examined how climate variability influences the distribution of malaria. They have indicated that highland malaria can respond to climatic variability, but whether changes in the altitudinal range of malaria which have apparently been observed in a number of sites are due to global climate change, is currently a matter of scientific debate. Only long-term monitoring of climate, vector populations, and the incidence of malaria, as well as potential confounding factors, such as changes in vector control programs and forest cover can finally resolve the controversy.

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